Black’s Model

Posted in Finance, Accounting and Economics Terms, Total Reads: 351
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Definition: Black’s Model

Fischer Black modified the Black-Scholes option pricing model in 1976 to create the Black’s Model where while valuing stock options, spot price of the underlying was replaced by discounted futures price. Hence, Black‘s model can be used to price derivatives (like swaptions and bond options) and capping the variable rate loans.


Also referred to as Black-76 model, it uses the fact that there are no financing costs in a futures contract, resulting in lower option price. There are several assumptions that are used while applying this model:

• Similar to Black Scholes model,

o Options are European style (exercised only on expiry date)

o The stocks (or any underlying instrument) does not pay dividends

o No commissions are charged

o Returns on the stock ( or any underlying instrument) are distributed in a lognormal fashion (hence, reducing price volatility)

o Risk free interest rate is taken as constant

• There are no margins or taxes levied


The Black formula is given as:

C = Ue-rTN(h) – Ee-rTN(h - vT1/2)

P = - Ue-rTN(-h) – Ee-rTN(vT1/2 - h)


Where,

h = {ln(U/E)}/ vT1/2 + (vT1/2)/2

C = value of call option

P = value of put option

U = underlying future contract price

E = exercise price

T = time left for expiry (in years)

v = annual volatility

r = risk free rate

e = natural logarithm base

ln = natural logarithm

N(x) = cumulative normal density function

 

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